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DSP Riddle: Infinitely Narrow Notch Filter

Started by Andor May 19, 2006
"Martin Eisenberg" <martin.eisenberg@udo.edu> wrote in message 
news:1148154198.209375@ostenberg.wh.uni-dortmund.de...
> Fred Marshall wrote: > >> I am speaking of discrete time / discrete frequency here. > > You can't meaningfully use the phrase "infinitely narrow" if you're > working on a grid. Alternatively, you might take the discrete > spectrum to specify a continuous spectrum through circular sinc > interpolation; but prescribing a zero at one grid point (and its > images) will leave the continuous notch finitely wide.
Martin, Why not? It seems I didn't make my point clearly enough. Make the grid arbitrarily fine. From a constructive point of view, it's the same thing as making the spacing between points approach being infinitely small seems to me. And the distance spanned by 3 points (athe distance from 1 to zero to 1) is only 2 times "infinitely narrow". How big is that? I see no contradiction in things being "infinitely small" vs. being infinite. Multiply by two, divide by two ... you still get things very small and very big. Fred
Martin Eisenberg wrote:

> Randy Yates wrote: > > >>So an infinitely-narrow spectrum is the same as no spectrum at all? > > > Yes, if it's bounded (if the flat hamburger has finite area).
It would be hard to bite the bun if it weren't. :-) Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Fred Marshall wrote:

> > "Martin Eisenberg" <martin.eisenberg@udo.edu> wrote in message > news:1148154198.209375@ostenberg.wh.uni-dortmund.de... >> Fred Marshall wrote: >> >>> I am speaking of discrete time / discrete frequency here. >> >> You can't meaningfully use the phrase "infinitely narrow" if you're >> working on a grid. Alternatively, you might take the discrete >> spectrum to specify a continuous spectrum through circular sinc >> interpolation; but prescribing a zero at one grid point (and its >> images) will leave the continuous notch finitely wide. > > Martin, > > Why not? It seems I didn't make my point clearly enough. > Make the grid arbitrarily fine. From a constructive point of view, it's > the same thing as making the spacing between points approach being > infinitely > small seems to me. And the distance spanned by 3 points (athe distance > from > 1 to zero to 1) is only 2 times "infinitely narrow". How big is that? > > I see no contradiction in things being "infinitely small" vs. being > infinite. Multiply by two, divide by two ... you still get things very > small and very big. > > Fred
Perhaps the point is that, no matter how fine the grid is, infinitesimally fine even, you still _sample_ at _countably_ many points (even if infinitely many). In order to approach continuous you would have to be able to "sample" at _uncountably_ many points. -- Jani Huhtanen Tampere University of Technology, Pori
"Jani Huhtanen" <jani.huhtanen@kolumbus.fi> wrote in message 
news:BBWbg.3522$eO3.2998@reader1.news.jippii.net...
> Fred Marshall wrote: > >> >> "Martin Eisenberg" <martin.eisenberg@udo.edu> wrote in message >> news:1148154198.209375@ostenberg.wh.uni-dortmund.de... >>> Fred Marshall wrote: >>> >>>> I am speaking of discrete time / discrete frequency here. >>> >>> You can't meaningfully use the phrase "infinitely narrow" if you're >>> working on a grid. Alternatively, you might take the discrete >>> spectrum to specify a continuous spectrum through circular sinc >>> interpolation; but prescribing a zero at one grid point (and its >>> images) will leave the continuous notch finitely wide. >> >> Martin, >> >> Why not? It seems I didn't make my point clearly enough. >> Make the grid arbitrarily fine. From a constructive point of view, it's >> the same thing as making the spacing between points approach being >> infinitely >> small seems to me. And the distance spanned by 3 points (athe distance >> from >> 1 to zero to 1) is only 2 times "infinitely narrow". How big is that? >> >> I see no contradiction in things being "infinitely small" vs. being >> infinite. Multiply by two, divide by two ... you still get things very >> small and very big. >> >> Fred > > Perhaps the point is that, no matter how fine the grid is, infinitesimally > fine even, you still _sample_ at _countably_ many points (even if > infinitely many). In order to approach continuous you would have to be > able > to "sample" at _uncountably_ many points. > > -- > Jani Huhtanen
Jani, OK. That's interesting. While perhaps helpful semantically, I don't think any of this is much to the point of my original post. I think we depart from the focus. To me, and others more articulate on the subject, mathematics is used to usefully describe "things" that we understand. Not exclusively perhaps but very importantly. Yes, sometimes the mathematics helps us to gain new insights. And, yes, we need good rules and terminology. I'm an engineer first, a physicist second and a mathematician third (if that). So, I think the construction of the description is useful. I am reminded of calculus developments where we say things like "in the limit" and that was behind my comments. As in: "as the sample rate approaches infinity in the limit". Perhaps what I said lacked formalism but that wasn't central to my purpose. The key question is: does a continuous function as the one we are contemplating have important characteristics that are different than one that is sampled at a very high rate - *for the purposes of the discussion at hand?* I think it doesn't. Now, I didn't say they were the same did I? What I suggest is that they lack important differences in the context of this chat. So, I believe the constructive approach is interesting and useful. Fred
Fred Marshall wrote:

> > "Jani Huhtanen" <jani.huhtanen@kolumbus.fi> wrote in message > news:BBWbg.3522$eO3.2998@reader1.news.jippii.net... >> Fred Marshall wrote: >> >>> >>> "Martin Eisenberg" <martin.eisenberg@udo.edu> wrote in message >>> news:1148154198.209375@ostenberg.wh.uni-dortmund.de... >>>> Fred Marshall wrote: >>>> >>>>> I am speaking of discrete time / discrete frequency here. >>>> >>>> You can't meaningfully use the phrase "infinitely narrow" if you're >>>> working on a grid. Alternatively, you might take the discrete >>>> spectrum to specify a continuous spectrum through circular sinc >>>> interpolation; but prescribing a zero at one grid point (and its >>>> images) will leave the continuous notch finitely wide. >>> >>> Martin, >>> >>> Why not? It seems I didn't make my point clearly enough. >>> Make the grid arbitrarily fine. From a constructive point of view, it's >>> the same thing as making the spacing between points approach being >>> infinitely >>> small seems to me. And the distance spanned by 3 points (athe distance >>> from >>> 1 to zero to 1) is only 2 times "infinitely narrow". How big is that? >>> >>> I see no contradiction in things being "infinitely small" vs. being >>> infinite. Multiply by two, divide by two ... you still get things very >>> small and very big. >>> >>> Fred >> >> Perhaps the point is that, no matter how fine the grid is, >> infinitesimally fine even, you still _sample_ at _countably_ many points >> (even if infinitely many). In order to approach continuous you would have >> to be able >> to "sample" at _uncountably_ many points. >> >> -- >> Jani Huhtanen >
> > The key question is: does a continuous function as the one we are > contemplating have important characteristics that are different than one > that is sampled at a very high rate - *for the purposes of the discussion > at > hand?* I think it doesn't.
Perhaps it doesn't. I'm not wise enough to say yes or no without spending some more time on this. However, one thing I have noticed, is that when unrealistic specifications with infinities are analysed, suprisingly unintuitive properties often appear. Just consider what happens to the step function when it is reconstructed through Fourier transform with increasing accuracy. The energy of the error gets smaller while the maximum error does not approach 0. This is known as Gibb's phenomenom (as you probably know).
> Now, I didn't say they were the same did I? What I suggest is that they > lack important differences in the context of this chat. So, I believe the > constructive approach is interesting and useful.
You didn't and I only tried to clarify Martin's point (although I'm not sure if this is what he was referring to). It is quite possible your construction is perfectly valid. However, in this case there is an "extreme" discontinuity and something similar to Gibb's might cause problems to your construction. -- Jani Huhtanen Tampere University of Technology, Pori
Fred Marshall wrote:

  ...

> I am reminded of calculus developments where we say things like "in the > limit" and that was behind my comments. As in: "as the sample rate > approaches infinity in the limit". Perhaps what I said lacked formalism but > that wasn't central to my purpose. ...
Limiting behavior is the key here to what happens when the poles approach ever closer to the zeros. If they sidle up subtly, there's an infinitely narrow notch. If they're set down plunk on top, there's pole-zero cancellation. It doesn't matter. An infinitely narrow sinusoid doesn't exist. There are sidebands from starting, even if we intend for it never to stop. It's as if every tone has a bit of vibrato, hence width, so an infinitely narrow filter not only takes forever to begin attenuating it, but it must let most of it pass even then. The nice part about playing with metaDSP is that one can get metaphysical without restraint or reproach. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jani Huhtanen wrote:

  ...

> Just consider what happens to the step function when it is reconstructed > through Fourier transform with increasing accuracy. The energy of the error > gets smaller while the maximum error does not approach 0. This is known as > Gibb's phenomenom (as you probably know). ...
Not only doesn't it approach zero, it doesn't shrink at all. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Friends,

as I wrote down my solution to the riddle, it turned out that the
solution was flawed! I guess we cannot have an infinitely narrow notch
after all. Here is what I wrote:


>Consider the linear transformation H which transforms > an input sequence x[n] into an output sequence y[n] > using the following transformation rule: > > y[n] = x[n] + c0 x[n-1] + x[n-2] - c0 y[n-1] - y[n-2] (1) > > with c0 = - 2 cos(w0).
>I will show the proposed properties of the system H directly: > >1. H(w) = 1, for w =/= w0, and 0 <= (w, w0) <= pi (thanks Randy).
[snip, because unimportant]
> 2. H(w0) = 0. > We look at the response of the system to the input sequence > x[n] = exp(j w0 n). For simplicity, we use an intermediate > variable u[n], and rewrite (1) as: > > u[n] = x[n] + c0 x[n-1] + x[n-2], (1.a) > y[n] = - c0 y[n-1] - y[n-2] + u[n]. (1.b) > > Computing u[n], we get: > u[n] = exp(j w0 n) + c0 exp(j w0 (n-1) ) + exp(j w0 (n-2)) > = exp(j w0 (n-1)) ( exp(j w0) + c0 + exp(-j w0) ) > = exp(j w0 (n-1)) ( 2 cos(w0) + c0) = 0 > > where in the last step we used c0 = -2 cos(w0). Since > u[n] = 0 for all n, so is y[n] (which follows from equation (1.b)).
That last comment in brackes is not correct. If u[n] = 0, then by equation (1.b) we have y[n] - 2 cow(w0) y[n-1] + y[n-2] = 0, which does _not_ imply y[n] = 0 for all n. In fact, the above is just the oscillator recursion, any y[n] = A1 exp(- j w0 n) + A2 exp(j w0 n), where A1 and A2 can be complex, is a solution to the recursion. I guess the problem is that no initial values of y[n] are defined, and therefore y[n] is not defined (just as 0 * oo). When we realize equation (1) for a causal sequence x[n], n >= 0, we will in general get an oscillation at w0 (depeding on the sequence x[n]). Interestingly, the amplitude and phase of the oscillation at w0 depends only on the first two values x[0] and x[1], and not on the rest of the sequence - these are the remains of the infinitely narrow notch. :-) Best regards, Andor
Andor wrote:

  ...

> y[n] - 2 cow(w0) y[n-1] + y[n-2] = 0, ...
Holy cow! A new function! :-) (See picture at http://tucows.com/.) Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jerry Avins wrote:
> Andor wrote: > > ... > > > y[n] - 2 cow(w0) y[n-1] + y[n-2] = 0, ... > > Holy cow! A new function! :-) (See picture at http://tucows.com/.)
Thanks Jerry - rub it in :-).